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Applied Combinatorics

Section 1.7 Sudoku Puzzles

Here’s an example which has more substance than you might think at first glance. It involves Sudoku puzzles, which have become immensely popular in recent years.

Example 1.21.

A Sudoku puzzle is a \(9\times 9\) array of cells that when completed have the integers \(1,2,\dots,9\) appearing exactly once in each row and each column. Also (and this is what makes the puzzles so fascinating), the numbers \(1\text{,}\) \(2\text{,}\) \(3,\dots,9\) appear once in each of the nine \(3\times 3\) subsquares identified by the darkened borders. To be considered a legitimate Sudoku puzzle, there should be a unique solution. In Figure 1.22, we show two Sudoku puzzles. The one on the right is fairly easy, and the one on the left is far more challenging.
Figure 1.22. Sudoku puzzles
There are many sources of Sudoku puzzles, and software that generates Sudoku puzzles and then allows you to play them with an attractive GUI is available for all operating systems we know anything about (although not recommend to play them during class!). Also, you can find Sudoku puzzles on the web at: On this site, the “Evil” ones are just that.
How does Rory make up good Sudoku puzzles, ones that are difficult for Mandy to solve? How could Mandy use a computer to solve puzzles that Rory has constructed? What makes some Sudoku puzzles easy and some of them hard?
The size of a Sudoku puzzle can be expanded in an obvious way, and many newspapers include a \(16\times16\) Sudoku puzzle in their Sunday edition (just next to a challenging crosswords puzzle). How difficult would it be to solve a \(1024\times1024\) Sudoku puzzle, even if you had access to a powerful computer?